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  • View in gallery

    Schematic representation of the isentropic shallow-water model with associated variables in (a),(b) the troposphere and (c),(d) in the stratosphere for (a),(c) a two-layer and (b),(d) a 1½-layer approximation . In the 1½-layer model the top layer is at rest [i.e., u1(x, y, t) ≈ 0] and is capped by a rigid lid [i.e., z(x, y, t) = Z0].

  • View in gallery

    Profiles, zonally averaged, (solid lines) of (a) observed potential temperature θ(z) and (b) pressure p(z) vs height z at circa 57.25°N, and extrapolated profiles (dash–dotted lines) from Z1 = 16.63 km to Z0 = 34.63 km based on the approximately constant scale heights of the observed θ and p, respectively, in the stratosphere. The tropopause lies at approximately Z2 = 10.63 km. Data courtesy Dr. Thomas Birner (cf. Birner 2006). The relevant physical parameters associated with these vertical profiles are reported in Table 1.

  • View in gallery

    Vertical profile of potential temperature (solid line) and wind speed (dashed line) taken from radiosonde data at (a) 0000 UTC 10 Dec 1977, (b) 1200 UTC 10 Dec 1977, and (c) 0000 UTC 11 Dec 1977 in Brownsville. The horizontal dotted lines indicate the depth of the two layers deduced from potential temperature data. The relevant physical parameters associated with each vertical profile are reported in Table 1. Source: http://weather.uwyo.edu/upperair/sounding.html.

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An Idealized 1½-Layer Isentropic Model with Convection and Precipitation for Satellite Data Assimilation Research. Part II: Model Derivation

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  • 1 aSchool of Mathematics, University of Leeds, Leeds, United Kingdom
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Abstract

In this Part II paper we present a fully consistent analytical derivation of the “dry” isentropic 1½-layer shallow-water model described and used in Part I of this study, with no convection and precipitation. The mathematical derivation presented here is based on a combined asymptotic and slaved Hamiltonian analysis, which is used to resolve an apparent inconsistency arising from the application of a rigid-lid approximation to an isentropic two-layer shallow-water model. Real observations based on radiosonde data are used to justify the scaling assumptions used throughout the paper, as well as in Part I. Eventually, a fully consistent isentropic 1½-layer model emerges from imposing fluid at rest (v1 = 0) and zero Montgomery potential (M1 = 0) in the upper layer of an isentropic two-layer model.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Onno Bokhove, o.bokhove@leeds.ac.uk; Luca Cantarello, mmlca@leeds.ac.uk

Abstract

In this Part II paper we present a fully consistent analytical derivation of the “dry” isentropic 1½-layer shallow-water model described and used in Part I of this study, with no convection and precipitation. The mathematical derivation presented here is based on a combined asymptotic and slaved Hamiltonian analysis, which is used to resolve an apparent inconsistency arising from the application of a rigid-lid approximation to an isentropic two-layer shallow-water model. Real observations based on radiosonde data are used to justify the scaling assumptions used throughout the paper, as well as in Part I. Eventually, a fully consistent isentropic 1½-layer model emerges from imposing fluid at rest (v1 = 0) and zero Montgomery potential (M1 = 0) in the upper layer of an isentropic two-layer model.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Onno Bokhove, o.bokhove@leeds.ac.uk; Luca Cantarello, mmlca@leeds.ac.uk

1. Introduction

In Cantarello et al. (2022, hereafter Part I), we presented and discussed both the dynamics and the numerics of a new idealized model (“ismodRSW”) to be used in future satellite data assimilation (DA) experiments. In this paper, or Part II, we show a formal mathematical derivation of the underlying isentropic 1½-layer shallow-water model based on variational principles and Hamiltonian fluid dynamics.

Shallow-water models represent a class of simplified fluid-dynamic models often utilized to describe analytically and numerically a number of fundamental and theoretical properties of stratified fluids, including the effect of rotation (e.g., as in the Rossby adjustment problem) and the propagation of gravity waves. In this regard, Zeitlin (2018) provides a broad overview of the use of shallow-water models in geophysics, including “moist” isentropic models able to mimic convection and precipitation. In recent decades, shallow-water models have also been utilized as idealized tools in DA research, for both oceanic and atmospheric applications (Žagar et al. 2004; Salman et al. 2006; Stewart et al. 2013; Würsch and Craig 2014; Kent et al. 2017).

Typically, shallow-water models emerge after vertically integrating the Navier–Stokes equations whenever the vertical motions can be neglected over wider zonal and meridional scales. The derivation of a simplified, isopycnal single-layer shallow-water model (i.e., a model with a single layer of fluid at constant density) is typical textbook material and can be found in many places (see, e.g., section 2 in the introduction of Zeitlin 2007). The derivation of multilayer shallow-water models is also covered extensively in many books (see, for instance, chapter 3 of Vallis 2017). One-and-a-half-layer models represent further simplifications in which the fluid is capped by a rigid lid and hence the total fluid depth is conserved in time. An isopycnal 1½-layer model differs from a single-layer model only in the definition of the gravity acceleration g and a reduced gravity g′ is introduced:
g=ρ2ρ1ρ2g,
in which ρ1 and ρ2 indicate the densities of the fluid in the upper and lower layer, with the least dense layer on top, i.e., ρ2 > ρ1. Instead, moving from an isopycnal model (constant density) to an isentropic one (constant potential temperature) leads to a different set of equations which, more importantly, are valid in a different atmospheric regime.

A “dry” isentropic 1½-layer shallow-water model (without convection and precipitation) should naturally arise from an isentropic two-layer model after imposing a rigid-lid condition on the top layer. Here, starting from the isentropic N = 2-layer model derived by Bokhove and Oliver (2009), we show that this approach leads to an apparent inconsistency in the model equations, in which a zero Montgomery potential constraint (M1 = 0) seems not to be preserved in time by the continuity equations of the layers. To resolve this contradiction, we adopt principles of Hamiltonian fluid dynamics (exploiting a slaved Hamiltonian approach) and introduce fast and slow variables (Van Kampen 1985) arising from an asymptotic analysis performed on an isentropic two-layer shallow-water model. Crucially, we will show that this asymptotic analysis relies on a series of scaling assumptions that can be justified on the basis of real-word observations obtained from radiosonde data, in the presence of low-level jet (LLJ) conditions. In the end, we show that the rigid-lid condition (M1 = 0) needs to be accompanied by fluid at rest in the top layer (v1 = 0) for the isentropic 1½-layer model to be a consistent approximation of a two-layer one.

The derivation of balanced fluid dynamical models exploiting Hamilton’s principle in which high-frequency waves are filtered out started with the work of Salmon (1983, 1985, 1988). In particular, the use of Dirac brackets’ theory (Dirac 1958, 1964) applied to the Hamiltonian derivation of multilayer shallow-water models was developed further in Bokhove (2002a) and Vanneste and Bokhove (2002). The derivation of an N-layer isentropic shallow-water model based on Hamiltonian mechanics was given in Bokhove and Oliver (2009) and will constitute the starting point of our study. In this regard, the reader might find useful to know that parts of the work treated in this paper has appeared in a previously unpublished manuscript (Bokhove 2007).

The structure of the paper is as follows. In section 2, we will start with presenting the equations of a full two-layer isentropic model and show how imposing a rigid-lid condition leads to a seemingly inconsistent yet closed 1½-layer model. In section 3 we introduce a scaling for the two-layer model and its equations are subsequently nondimensionalized; afterward, an asymptotic analysis based on the method of multiple time scales is conducted. In section 4 we use radiosonde observations to justify the scaling used in the asymptotic analysis. In section 5, the Hamiltonian derivation of the isentropic 1½-layer shallow-water model is discussed. Conclusions are given in section 6.

2. A rigid-lid approximation in a two-layer model

We start this section by presenting an isentropic two-layer model and finish with an argument how a closed 1½-layer model emerges by taking a seemingly inconsistent rigid-lid approximation. That the final model is nonetheless consistent will be subsequently shown in a combined asymptotic and Hamiltonian analysis, resulting in a rigid-lid condition with a (nearly) passive and high upper layer. The Hamiltonian derivation demonstrates that the 1½-layer model has a bona fide conservative and hyperbolic structure, which is exploited in the numerical discretization discussed in Part I.

A full, geometric derivation of an isentropic N–layer model can be found in Bokhove and Oliver (2009). Here, we take a two-layer simplification thereof, with N = 2. Figures 1a and 1c provide a sketch of the two-layer model configuration. The momentum equations of the model arise by assuming hydrostatic balance and constant entropy (potential temperature θ) in each layer. The continuity equations emerge once the space (x, y) and time-dependent (t) pseudodensity σα(x, y, t) for each layer, numbered by α = 1, 2, is defined, i.e.,
σα=pr(ηαηα1)/g,
in which g refers to the gravity acceleration and ηαηα−1 is the net nondimensional pressure difference between the bottom and the top of the layer α, with η defined as η = p/pr for a reference pressure pr. The pseudodensity σ arises from hydrostatic balance dp = −ρgdz, integrating an element of mass flux for some infinitesimal surface element dA: dm/dA = ρdz = −dp/g across each layer with density ρ, pressure p, and the gravitational acceleration g (note that pressure and density vary throughout the layer). In Bokhove (2002b) and Ripa (1993) the variational and Hamiltonian formulation of the isentropic N-layer equations are derived by simplifying the Eulerian variational principle of the compressible Euler equations.
Fig. 1.
Fig. 1.

Schematic representation of the isentropic shallow-water model with associated variables in (a),(b) the troposphere and (c),(d) in the stratosphere for (a),(c) a two-layer and (b),(d) a 1½-layer approximation . In the 1½-layer model the top layer is at rest [i.e., u1(x, y, t) ≈ 0] and is capped by a rigid lid [i.e., z(x, y, t) = Z0].

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0023.1

The resulting four, isentropic two-layer (continuity and momentum) equations are the following:
tσα+·(σαvα)=0,
tvα+(vα·)vα+fvα=Mα,
with α = 1,2 and in which is the horizontal gradient, vα=vα(x,y,t)=(uα,υα)T is the horizontal velocity within layer α, and vα=(υα,uα)T the vector perpendicular to it, f is the Coriolis frequency, and Mα is the Montgomery potential. To close the system, one needs to specify the Montgomery potentials in each layer. As seen in section 3 of Bokhove and Oliver (2009), for a two-layer model these potentials can be defined as
M1=cpθ2η2κ+cp(θ1θ2)η1κ+gz2,
M2=cpθ2η2κ+gz2,
in which κ = R/cp is the ratio between the specific gas constant for dry air (R = 287 J kg−1 K−1) and its specific heat capacity at constant pressure (cp = 1004 J kg−1 K−1).
The hydrostatic condition for an isentropic model ∂M/∂z = 0 implies that, in general, the Montgomery potential M = cpθηκ + gz is independent of z within each layer. Therefore, one can evaluate M in the bottom layer (where θ = θ2) at both z = z2 and z = z1, and M in the upper layer (where θ = θ1) at both z = z1 and z = z0, to find
gz0=cpθ1(η1κη0κ)+gz1,
gz1=cpθ2(η2κη1κ)+gz2,
from which is possible to express the thickness of each layer as
h1=z0z1=(cpθ1/g)(η1κη0κ),
h2=z1z2=(cpθ2/g)(η2κη1κ).
We note here that the nondimensional pressure η0 is treated as a constant throughout the paper.
Finally, the relations between layer pressure and pseudodensities can be derived using the expressions (2) for σ1 and σ2 as follows:
η1=gσ1/pr+η0andη2=g(σ1+σ2)/pr+η0.
When one takes a rigid-lid approximation, it is convenient to add a constant K=(cpθ1η0κ+gZ0) to M1 in (3c), leading to
M1=cpθ1(η1κη0κ)+cpθ2(η2κη1κ)+gz2gZ0.
Therefore, by substituting (4b) into (4a) and subtracting gZ0 from both sides one finds
gz0=cpθ1(η1κη0κ)+cpθ2(η2κη1κ)+gz2,gz0gZ0=cpθ1(η1κη0κ)+cpθ2(η2κη1κ)+gz2gZ0,gz0gZ0=M1.

If the top surface is fixed, i.e., z0 = Z0, then M1 = g(z0Z0) = 0, and a closed 1½-layer model emerges as follows (a sketch of the model is given in Figs. 1b,d). For the 1½-layer model, the momentum equations in the lower stratospheric layer remain as in Eqs. (3a) and (3b) for α = 2. The model is indeed closed, because M1 = 0 defines η1 = p1/pr in terms of η2 = p2/pr. This fact allows σ2 to be expressed in terms of η2 as follows: σ2(η2) = pr[η2η1(η2)]/g. We note that such a 1½-layer model has the advantage over a one-layer model that the pressure p1 is active and not constrained to be constant, as is p0. Consequently, the values of the surface pressure p2 are more realistic. At first sight, the 1½-layer model, however, seems inconsistent, since the constraint M1 = 0 is not preserved in time by the original two continuity equations. Nevertheless—as we will show later in this paper—the closed 1½-layer model [(3a) and (3b)] with α = 2 and Montgomery potential M2 results after taking M1 = 0 and v1 = 0 in the momentum equation of the stratospheric layer. Perhaps not surprisingly, the original potential energy of the two-layer model subject to the constraint M1 = g(z0Z0) = 0 does give the desired potential energy of the 1½-layer model.

3. Scaling of a two-layer model and asymptotic analysis

a. Nondimensionalization and scaling of the two-layer model

To perform asymptotic analysis on the two-layer model, we first nondimensionalize Eqs. (3) by applying the following scaling:
(x,y)=L(x*,y*),t=(L/U2)t*,vα=Uαvα*,Mα=gHαMα*,=(1/L)*,σ1=(pr/g)σ1*=(pr/g)(Σ1+ε2σ1),σ2=ε2(pr/g)σ2*,pα=prηα,θα=(gH1/cp)θα*,hα=Hαhα*,Z0=H1Z0*,z2=Fr22H2z2*,
together with the following scaling approximations:
Fr1ε,andδaFr22ε2,
in which L is the horizontal length scale; Uα and Hα are the layer velocity and depth scale, respectively; Σ1 is a constant; Frα indicates the layer Froude number Frα=Uα/gHα; cpθ1/gH1 is assumed to be O(1); ε is the layer velocity ratio ε = U1/U2; δa = H2/H1 defines the layer thickness ratio. Both ε and δa are assumed to be small and the assumption made in (9) implies a scaling on δa via the Froude number. We will discuss in section 4 to what extent the chosen scaling is supported by observations.
After dropping the asterisks, the scaling (8) substituted in (5) yields
h1=cpgH1gH1cpθ1(η1κη0κ)=θ1(η1κη0κ)=θ1(σ1κη0κ),
h2=cpgH2gH1cpθ2(η2κη1κ)=θ2δa(η2κη1κ)=θ2δa[(σ1+ε2σ2)κσ1κ],
in which we use (6), and make the assumption that η0 is small compared to (nondimensional) σ1. Again, the validity of the latter on observational evidence is discussed in section 4; nevertheless, even in absence of such hypothesis, the rest of the derivation would only differ for the presence of an additional constant.
By substituting (8) and (9) further into (3), and retaining the asterisks only for the Montgomery potentials, one obtains
tσ1+ε·(σ1v1)+1ε·(Σ1v1)=0,
tv1+ε(v1·v1+1Ro1v1)+1ε3M1*=0,
tσ2+·(σ2v2)=0,
tv2+v2·v2+1Ro2v2+1Fr22M2*=0,
with Roα being the layer Rossby number Roα=Uα/(fL) and with M1* and M2* being
M1*=θ2*η2κ+(θ1*θ2*)η1κθ1*η0κZ0*+ε2z2*,
M2*=1δaθ2*η2κ+Fr22z2*,
after adding the constant K=(cpθ1η0κ+gZ0) to M1. Therefore, we define the (constant) mean potentials M¯1 and M¯2:
M¯1=θ1*Σ1κθ1*η0κZ0*,
M¯2=1δaθ2*Σ1κ,
which can be subsequently subtracted from M1* and M2* in (12), as it is always possible to vary a potential by a constant without loss of generality. Finally, by bringing a factor 1/ε2 inside M1* and a factor 1/Fr22 inside M2* in (12), we obtain the quantities
M1=(M1*M¯1)/ε2,
M2=(M2*M¯2)/Fr22.
Hence, the system of equations in (11) can be rewritten as
tσ1+ε·(σ1v1)+1ε·(Σ1v1)=0,
tv1+ε(v1·v1+1Ro1v1)+1εM1=0,
tσ2+·(σ2v2)=0,
tv2+v2·v2+1Ro2v2+M2=0,
with potentials M1 and M2 defined as
M1=θ1[(Σ1+ε2σ1)κ(Σ1)κ]/ε2+θ2[(Σ1+ε2σ1+ε2σ2)κ(Σ1+ε2σ1)κ]/ε2+z2,M2=θ2[(Σ1+ε2σ1+ε2σ2)κ(Σ1)κ]/ε2+z2,
in which the scaling assumption for δa in (9) has been used.

As will appear clearer in both the asymptotic analysis and the Hamiltonian derivation, the mean potentials M¯1 and M¯2 are chosen primarily to avoid singularities at leading order in ε.

b. Asymptotic analysis

Asymptotic analysis of the upper layer at leading order in ε—that is, at O(1/ε) in (15)—yields two constraints:
ϕ1=M1|ε=0=0andD1=·(Σ1v1)=0,
with leading-order potentials (obtained by computing the Taylor expansions of (16) around σ1=0, σ2 = 0):
M1|ε=0=κΣ1κ1(θ1σ1+θ2σ2)+z2andM2|ε=0=κθ2(Σ1)κ1(σ1+σ2)+z2.

We introduce a fast time scale τ = t/ε and evaluate (15) at leading order; that is, we truncate the system (15) and (16) at the fast time scale by taking the limit ε → 0 (after multiplication by ε).

The following linear wave equations then appear after some manipulation:
τσ1+D1=0,τD1=·(Σ1M1|ε=0),τω1=0,τσ2=0,τv2=0,
with vorticity ω1=·v1. The system (19) shows that the fast variables σ1 and D1 oscillate rapidly, while the slow variables ω1, σ2, and v2 vary on the slow time scale. The introduction of fast and slow variables is based on the distinction between high-frequency and low-frequency waves in linearized wave equations (Van Kampen 1985). Later in section 5, we will consider the reduced, Hamiltonian dynamics on the “slow” manifold defined by the constraints (17).

4. Observations supporting the scaling

The validity of the scaling used in section 3 determines whether the 1½-layer model to be derived in this paper is suitable to represent the real atmosphere. In this sense, here we show two possible applications: one in the stratosphere and one in the troposphere, with the latter being more relevant for the purpose of Part I of this study, as convection and precipitation are confined therein. Table 1 summarizes the values derived from real atmospheric measurements (i.e., from radiosonde data) of the main relevant physical quantities together with the associated nondimensional parameters Fr1obs, Fr2obs, δa, and ε. A sketch of the model configuration for both cases is shown in Fig. 1.

Table 1

Summary of the values of various physical quantities obtained from the radiosonde data displayed in Figs. 2 and 3 and resulting values of nondimensional scaling parameters δa and ε. The values of Fr1 and Fr2—scaling hypotheses made earlier in section 3—are also reported. The values of p1 and θ1 in the bottom rows are computed via (5) using the observed quantities p0obs, p2obs, H1obs, H2obs, and θ2obs. The rightmost column reports the average values obtained from the data seen in Fig. 3.

Table 1

a. Two-layer stratosphere

The scaling is compatible with a 1½-layer approximation of the stratosphere. In this regard, our estimates are based on zonally averaged climatological seasonal radiosonde data displayed in Birner (2006), where potential temperature and horizontal wind speed are displayed as function of height and latitude. On request, Dr. Birner extracted vertical profiles of potential temperature and pressure at about 57.25°N (i.e., at midlatitudes) versus height in the summer season, shown in Fig. 2. From the data, it seems reasonable to take Z2 = 10.6 km for the tropopause height (the lower bound) and, say, Z1 = 16.6 km and Z0 = 34.6 km. Hence, H2obs=6km and H1obs=18km. Estimates for the horizontal velocities are U1obs2m s1 and U2obs14m s1 [from Fig. 7 of Birner (2006) and T. Birner 2007 and 2021, personal communication]. From Fig. 2, average values of the potential temperature are found to be approximately θ2obs=381K and θ1obs=672K. Likewise, pressures observed and deduced at these heights are p2obs=242mb, p1obs=97mb, and p0obs=6.2mb, and therefore η0 M σ1. In our scaling, Fr1obs, Fr2obs, ε, δa, θ1, and p1 follow, for example, after choosing θ2obs, H1obs, H2obs, p2obs, p0obs, U1obs, and U2obs, and exploiting Eqs. (5). Further constants used are, g = 9.81 ms−1, cp = 1004.6 J kg−1 K−1, R = 287.04 J kg−1 K−1, and pr = 1000 mb such that κ = 2/7. We obtain, Fr1obs0.0048, Fr2obs0.0577, ε ≈ 0.14, δa ≈ 0.33, θ1 = 629 K [solving (5a) for θ1], and p1 = 97 mb [solving (5b) for η1 and hence p1]. All data are reported in Table 1. This pressure value compares well with the observed one, while the calculated and observed potential temperature differ somewhat because the observed buoyancy frequency (and temperature) is roughly constant and not the entropy as in our layer model. The values of the Froude numbers Fr1 and Fr2 deduced from the scaling parameters ε and δa are within a factor 5 from those deduced from the observations Fr1obs and Fr2obs. For the above values, cpθ1/(gH1) = 3.58. Despite these slight differences, the data provide an observational basis for the chosen scaling.

Fig. 2.
Fig. 2.

Profiles, zonally averaged, (solid lines) of (a) observed potential temperature θ(z) and (b) pressure p(z) vs height z at circa 57.25°N, and extrapolated profiles (dash–dotted lines) from Z1 = 16.63 km to Z0 = 34.63 km based on the approximately constant scale heights of the observed θ and p, respectively, in the stratosphere. The tropopause lies at approximately Z2 = 10.63 km. Data courtesy Dr. Thomas Birner (cf. Birner 2006). The relevant physical parameters associated with these vertical profiles are reported in Table 1.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0023.1

b. Two-layer troposphere in presence of a low-level jet

The scaling presented in section 3 is also compatible with a 1½-layer approximation in the troposphere. LLJs are recurrent meteorological features located at various locations in the world (Rife et al. 2010) and they happen to be particularly common over the Great Plains in the southern United States (Ladwig 1980; Djurić and Damiani 1980).

Figure 3 shows vertical profiles obtained from radiosonde data of both potential temperature and wind speed during an LLJ event on 10–11 December 1977 in Brownsville, Texas (United States). We use this as a case study to provide a justification for the scaling chosen in section 3. We approximate the troposphere as a two-layer fluid, exploiting the discontinuities in the potential temperature profile of Fig. 3b as a reference. Mean potential temperature values of θ1obs=311.0K and θ2obs=291.8K follow after taking H1obs=4.02km and H2obs=2.08km in Fig. 3b. The above values of θ1obs and θ2obs are used as a constraint to compute H1obs and H2obs also in the profiles of Figs. 3a–c, in virtue of the isentropic assumption (i.e., constant potential temperature within each layer). Once layer depths in each profile are established, mean wind speed values U1obs and U2obs within each layer are also computed (dashed line in Fig. 3). Table 1 summarizes all the other relevant physical parameters associated with the radiosonde data plotted in Fig. 3, including the values of pressure p0obs, p1obs, and p2obs obtained for each profile (vertical profiles of pressure not shown). In this case, η0 > σ1. All in all, it is possible to see from Table 1 how ε and δa lie below one during the LLJ event; moreover, the rigid-lid condition leading to the 1½-layer configuration appears to be justified, as the variation in height of Z0 = H1 + H2 = 6, 6.1, and 6.25 km is smaller than the change in depth of the bottom layer H2 = 2.02, 2.08, and 1.65 km. Furthermore, the values of p1 and θ1 computed via Eq. (5) using the observed values θ2obs, H1obs, H2obs, p2obs, and p0obs are very close to p1obs and θ1obs coming from the observations themselves. The values of the Froude numbers Fr1 and Fr2 deduced from the scaling parameters ε and δa are on average (cf. rightmost column in Table 1) within a factor of 8 from those deduced from the observations Fr1obs and Fr2obs. Finally, cpθ1obs/(gH1obs){8.00,7.92,6.92} for the three vertical profiles of Fig. 3. Overall, the data support the scaling (8) chosen in section 3.

Fig. 3.
Fig. 3.

Vertical profile of potential temperature (solid line) and wind speed (dashed line) taken from radiosonde data at (a) 0000 UTC 10 Dec 1977, (b) 1200 UTC 10 Dec 1977, and (c) 0000 UTC 11 Dec 1977 in Brownsville. The horizontal dotted lines indicate the depth of the two layers deduced from potential temperature data. The relevant physical parameters associated with each vertical profile are reported in Table 1. Source: http://weather.uwyo.edu/upperair/sounding.html.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0023.1

5. Hamiltonian derivation

a. Constrained Hamiltonian formulation of the 1½-layer equations

A dimensional Hamiltonian formulation of the two-layer system is introduced to derive the formulation for the 1½-layer system. It consists of the evolution
dFdt={F,H},
with the shallow-layer generalized Poisson bracket in both layers (α = 1, 2):
{F,G}=α=12[qα(δFδvα)·δGδvαδFδσα·δGδvα+δGδσα·δFδvα]dxdy,
for arbitrary functionals F and G of {vα,σα} and a Hamiltonian of the type
H=[α=12(12σα|vα|2+gσαz2)+prcpθ2g(κ+1)(η2κ+1η1κ+1)+prcpθ1g(κ+1)η1κ+1σ1(cpθ1η0κ+gZ0)]dxdy,
and the potential vorticity qα in each layer α:
qα=(f+·vα)/σα,
appearing in (21). The bracket in (21) is the same as that in Bokhove (2002b). The Hamiltonian follows either directly from the Eulerian or parcel Eulerian–Lagrangian momentum equations or from the Hamiltonian of the 3D Euler equations by neglecting the vertical velocity relative to the horizontal velocities, by using hydrostatic balance and the ideal gas law, and integration in the vertical over each isentropic layer. In the latter integration, the horizontal velocity is assumed to be independent of the depth in each layer and the last term in (22) is then absent, whereupon the resulting Hamiltonian matches (3.16) in Bokhove (2002b) after noting that the layer numbers are the same but the levels are numbered starting from 1 at the top. This last term, linear in σ1, arises without problem because any multiple of the mass σ1dxdy in the upper layer is a Casimir invariant and can be added to the Hamiltonian without changing the dynamics (cf. Shepherd 1990). After taking the variation, it amounts to adding a constant to the Montgomery potential, which is always allowed. This addition further ensures that M1 = g(z0Z0), which is a useful simplification as we have seen.
The functional derivatives of the Hamiltonian (22) are
δHδvα=σαvαandδHδσα=|vα|2/2+Mα,
with which it can be verified that (20) yields the equations of motion (3) in both layers when we choose the functionals:
F=vα(x,t)=δ(xx)vα(x,t)dxdyandF=σα(x,t)=δ(xx)σα(x,t)dxdy,
respectively [see, e.g., Shepherd (1990) and Salmon (1988) for an introduction on Hamiltonian fluid dynamics], in which the first expression can be separated into two scalar quantities defined for each velocity component vα = (uα, υα). The second part of (24) follows from (22) after some calculation using the definitions of the Montgomery potentials in (3c) and (3d) and the pseudodensity (6) in (3).
The formulation (20)(22) is Hamiltonian as the bracket {F,G} is antisymmetric (i.e., {F,G}={G,F}) and satisfies the Jacobi identity, i.e.,
{F,{G,K}}+{G,{K,F}}+{K,{F,G}}=0
for arbitrary functionals F, G, and K.

In the verification of the properties above (as well as in some others later in the paper), certain integral terms vanish in presence of certain boundary conditions such as periodic boundaries; quiescence and constancy at infinity where σα is constant and vα = 0; slip flow along walls, such that vα·n^=0 with n^ the outward-pointing normal to the wall; or a combinations of these. Henceforth in this section, we use for simplicity periodic boundary conditions or quiescence and constancy at infinity. Furthermore, in these verifications the functional derivatives have to be restricted to satisfy corresponding boundary conditions.

Once that the Hamiltonian formulation for the two-layer model is established, we can use (8) to scale the Hamiltonian dynamics as follows:
dFdt={F,H}=[εq1(δFδv1)·δHδv11εδFδσ1·δHδv1+1εδHδσ1·δFδv1+q2(δFδv2)·δHδv2δFδσ2·δHδv2+δHδσ2·δFδv2]dxdy,
in which we have scaled the Hamiltonian with prU12L2/g and the functional derivatives with 1/L2, with potential vorticities:
q1=(1/Ro1+ω1)/(Σ1+ε2σ1)andq2=(1/Ro2+·v2)/σ2,
and modified Hamiltonian:
H={12(Σ1+ε2σ1)|v1|2+σ1z2+12σ2|v2|2+σ2z2+1ε4θ2κ+1[(Σ1+ε2σ1+ε2σ2)κ+1(Σ1+ε2σ1)κ+1]1ε4θ1(Σ1)κ+1+θ1ε41κ+1(Σ1+ε2σ1)κ+1(θ1σ1+θ2σ2)ε2(Σ1)κ}dxdy.
The additional terms, constant and linear in σ1 and σ2, are added to obtain a Hamiltonian of O(1) that is nonsingular as ε → 0. These extra terms arise because mass is globally conserved in each layer and can be introduced formally by adding constants and mass Casimirs C1=λ1(Σ1+ε2σ1)dxdy and C2=λ2ε2σ2dxdy to the original, scaled Hamiltonian H˜ for suitable choices of λ1 and λ2 (more details are given in appendix A). These above Casimirs are conserved since dC1/dt={C1,H˜}=0 and dC2/dt={C2,H˜}=0. The above expression is related but not quite equivalent to the available potential energy (Shepherd 1993). Here it suffices to note that it yields the proper equations of motion. Akin to the dimensional case, the variational derivatives of (28) are readily calculated to be
δHδσ1=ε2|v1|2/2+M1,δHδσ2=|v2|2/2+M2,δHδv1=(Σ1+ε2σ1)v1,andδHδv2=σ2v2.
Finally, the substitution of (29) into (26) yields the scaled equations of motion (15) with (16).
Given the constraints φ1 = M1 = 0 and D1=·(Σ1v1)=0, we can transform the generalized Poisson bracket, (26), in terms of the six variables (vα,σ1,σ2) to the variables (ϕ1,D1,ω1,v2,σ2) with ω1=·v1 being the vorticity in the top layer. The functional derivatives with respect to the former variables relate to those in terms of the latter variables as follows (see appendix B):
δFδv1|σ1=(δFδω1+Σ1δFδD1),δFδv2|σ1=δFδv2|ϕ1,δFδσ1|v1=M1σ1δFδϕ1,δFδσ2|v1,σ1=δFδσ2|ϕ1+M1σ2δFδϕ1,
in which the subscripts on the left-hand sides are used to avoid confusion on which set of variables is considered. After substitution of (30) into (26) and some rearrangement, we find
dFdt={F,H}={εq1J(δFδω1,δHδω1)+ε(Σ1)2q1J(δFδD1,δHδD1)+εΣ1q1[(δHδω1)·δFδD1(δFδω1)·δHδD1]+1εM1σ1[δFδϕ1·(Σ1δHδD1)δHδϕ1·(Σ1δFδD1)]+q2δFδv2·δHδv2(δFδσ2+M1σ2δFδϕ1)·δHδv2+(δHδσ2+M1σ2δHδϕ1)·δFδv2}dxdy,
with J(a, b) := (∂xa)(∂yb) − (∂xb)(∂ya) being the Jacobian operator. Note that, from (27), it follows that qα is O(1).
From (31) we derive the following system of equations for the new set of variables (ϕ1, D1, ω1, v2, σ2):
ϕ1(x,y,t)t={ϕ1(x,y,t),H}=1εM1σ1·(Σ1δHδD1)M1σ2·δHδv2,D1(x,y,t)t={D1(x,y,t),H}=ε·(Σ1q1δHδω1)+εJ(δHδD1,Σ12q1)1ε·[Σ1(M1σ1δHδϕ1)],ω1(x,y,t)t=εJ(q1,δHδω1)+ε·(Σ1q1δHδD1),v2(x,y,t)t=q2δHδv2(δHδσ2+M1σ2δHδϕ1),σ2(x,y,t)t=·δHδv2.
At leading order in ε the variational derivative of the Hamiltonian, (28), is (see appendix C):
δH|ε=0=[χδD1Ψδω1+M1|ε=0δσ1+σ2v2·δv2+(12|v2|2+M2|ε=0)δσ2]dxdy,
in which we have used Σ1v1=Σ1χ+Ψ with velocity potential χ and (transport) streamfunction Ψ. Therefore, using (33) one finds
δHδD1|ε=0=χandδHδσ1|ε=0=M1|ε=0.
By evaluating (32) at leading order in ε one obtains
εϕ1tD1=0andεD1t·(Σ1M1|ε=0)=0,
producing the constraints (17) as a solution, which shows consistency at leading order. Hence, at leading order in ε we take δH/δϕ1|ε=0=δH/δD1|ε=0=0 and from (32) we find the balanced dynamics on the slow manifold; that is, we truncate the dynamics to the leading-order terms in ε.
First, the vorticity dynamics in the upper layer is frozen in time:
tω1=0,
which we further simplify by initializing ω1(x, y, 0) = 0. Together with D1 = 0, this explains why it is asymptotically allowed to take v1 = 0 at leading order, as we discussed at the end of section 2.
Second, the balanced dynamics in the lower layer then becomes
v2t=q2δH0δv2δH0δσ2,andσ2t=·δH0δv2,
with H0 arising from (28) as the leading-order Hamiltonian on the constrained manifold [cf. (A5) with v1 = 0]:
H0={12σ2|v2|2+(σ1+σ2)z2+12θ2κΣ1κ1[(σ1+σ2)2σ22]+12θ1κΣ1κ1σ12}dxdy.
Variation of (38) gives [cf. (C1) with M1|ε=0=0]
δH0=[σ2v2·δv2+(12|v2|2+M2|ε=0)δσ2+M1|ε=0δσ1]dxdy,=[σ2v2·δv2+(12|v2|2+M2|ε=0)δσ2]dxdy,
using the constraint M1|ε=0=0, see Eqs. (17) and (18). Alternatively, by including higher-order terms in ε and using the (higher-order) constraint M1=0 in the Hamiltonian, we can use the original Hamiltonian (28) on the constrained manifold v1 = 0 [by initializing ω1(x, y, 0) = 0] and M1=0. The generalized Poisson bracket is then truncated to leading order on the (leading-order) constrained manifold, but the Hamiltonian Hv1=0,M1=0 includes higher-order terms in ε. When we truncate this higher-order Hamiltonian one finds again H0, of course, as Hv1=0,M1=0ε0H0 with M1|ε=0=0. This reduced Hamiltonian is chosen because it simply amounts to setting v1 = 0 and z0 = Z0 to get the rigid-lid approximation M1=0 in the Hamiltonian, which provides a physical procedure for our approximation.
The dynamics on the constrained manifold is governed by the slow variables {ω = 0, v2, σ2}, since the dynamics of the fast variables {D1, σ1} or {D1, φ1} associated with the gravity waves in the top layer is absent at leading order. Restricting or truncating the transformed bracket (31) to the constrained manifold and keeping all leading-order terms in ε, the following (dimensional and dimensionless) constrained dynamics emerges:
dFcdt={Fc,Hc}c=[q2δFcδv2·δHcδv2δFcδσ2·δHcδv2+δHcδσ2·δFcδv2]dxdy,
with the constrained Hamiltonian either Hc=H0 (with M1|ε=0=0) or Hc=Hv1=0,M1=0. We emphasize that Fc and Hc are functionals of the slow variables v2 and σ2 only.

b. The Jacobi identity

The bracket (40) satisfies the Jacobi identity since it coincides with the bottom-layer terms in the original bracket, (26), which consists of two uncoupled parts, one for each layer, to which the Jacobi identity can be applied separately. In this sense, the preservation of the Jacobi identity for the leading-order reduced bracket, (40) is straightforward to prove in the asymptotic analysis presented in this paper. However, proving the Jacobi identity for the leading-order reduced bracket resulting from a singular perturbation approach in the general case is more complicated and we refer to Bokhove (1996, 2002a) for a more extensive discussion of this topic.

c. Dimensional dynamics

Finally, the dynamics on the constrained manifold is given by (20) for F=Fc and H=Hc with (40) and the dimensional constrained Hamiltonian
Hc=[12σ2|v2|2+g(σ1+σ2)z2+cpprθ2g(κ+1)(η2κ+1η1κ+1)+cpprθ1g(κ+1)η1κ+1σ1(cpθ1η0κ+gZ0)]dxdy,
with σ2 = (p2p1)/g, σ1 = p1/g, and the constraint M1 = 0 (i.e., z0 = Z0) relating η1 = p1/pr to η2 = p2/pr, that is,
M1=cpθ2η2κ+cp(θ1θ2)η1κ+g(z2Z0)=0.
As argued earlier, instead of using the constrained Hamiltonian truncated to leading order in ε, we use the original Hamiltonian reduced to the constraint, or “rigid-lid” manifold, M1 = 0 (and v1 = 0). Hence, we include higher-order terms in ε in the Hamiltonian. This does not hamper the leading-order accuracy since the constrained bracket (40) is leading order. The functional derivative of the potential and internal energy in (41) subject to constraint (42) is
δHciδσ2δσ2=(z2+cpθ2gη2κ)δp2+(cp(θ1θ2)gη1κZ0)δp1=(z2+cpθ2gη2κ)δp2+(cp(θ1θ2)gη1κZ0)p1p2δp2=(z2+cpθ2gη2κ)(1p1p2)δp2=M2δσ2,
using the definition 2 = p2p1 and with Hci(σ2)=Hc(v2=0,σ2) denoting the nonkinetic terms in the Hamiltonian. The equations of motion (3) for α = 2 thus stay the same with Montgomery potential (3c), in which σ1 is defined in terms of σ2 and z2 by M1 = 0 via (3d).

Recapitulating, we note that we have been able to construct the Hamiltonian formulation of an isentropic 1½-layer model. Importantly, we conclude a posteriori that it is consistent to set v1 = 0, since in the upper layer we found ω1 = 0 by initializing ω1(x, y, 0) = 0 and D1 = 0 in the small ε limit.

6. Conclusions

In this paper, Part II, we have provided a full mathematical derivation of the isentropic 1½-layer shallow-water model utilized in Part I. Starting from an isentropic two-layer model, we show how a rigid-lid constraint alone (leading to the condition on the Montgomery potential in the top layer M1 = 0) does not suffice to derive an entirely consistent 1½-layer model, resulting instead in an apparent inconsistent configuration due to the nonpreservation in time of the M1 = 0 constraint.

To resolve this apparent inconsistency, we have shown how the 1½-layer model emerges from the two-layer one, once the latter is properly scaled to allow for the asymptotic analysis. In the limit ε = U1/U2 → 0, two constraints emerge, i.e., M1 = 0 and v1 = 0, and the system reduces to a single set of equations for the (slow) variables in the bottom layer.

We have further demonstrated that the scaling used in the asymptotic analysis can be justified on the basis of real observations; these arise in both the modeling of a two-layer stratosphere and that of a two-layer troposphere in the presence of a low-level jet. The latter is the most useful in view of using the idealized model described in Part I for satellite data assimilation research.

Finally, a Hamiltonian derivation of the model has been undertaken in section 5, where a slaved Hamiltonian approach has been used—generalized here for the infinite-dimensional case—thus removing an apparent inconsistency in the model derivation as well as underpinning the conservative nature of the system.

Acknowledgments.

This work stems in part from the work done by Luca Cantarello under a NERC SPHERES DTP scholarship (NE/L002574/1, Reference 1925512), cofunded by the Met Office via a CASE partnership. We thank the Deputy Chief Meteorologist Nicholas Silkstone (Met Office) for his useful suggestion regarding the low-level jet conditions that has guided our choice on a suitable scaling for our model in the troposphere. We also thank Gordon Inverarity (Met Office), Prof. Rupert Klein (Freie Universität Berlin), and another anonymous reviewer for their constructive comments. Finally, our thanks go to Prof. Thomas Birner for the discussion and use of his data (2006/07). We do not have conflicts of interest to disclose.

Data availability statement.

The stratospheric observational data used in section 4 can be found in the referenced paper (Birner 2006), while the tropospheric ones are obtained from the University of Wyoming’s Atmospheric Soundings web page (see http://weather.uwyo.edu/upperair/sounding.html).

APPENDIX A

Scaled Hamiltonian of a Two-Layer Shallow-Water Model

The scaled Hamiltonian displayed in Eq. (28) can be obtained by multiplying the dimensional two-layer Hamiltonian in (22) by a factor g/(prU22ε2L2) and by applying the scaling in (8). This leads to the expression
H=[12(Σ1+ε2σ1)|v1|2+1ε2Σ1z2+σ1z2+12σ2|v2|2+σ2z2+1ε4(κ+1)θ2(η2κ+1η1κ+1)+1ε4(κ+1)θ1η1κ+11ε4(θ1η0κ+Z0)(Σ1+ε2σ1)]dxdy,
which can be conveniently modified by adding the Casimirs invariants:
C1=λ1(Σ1+ε2σ1)dxdy,
C2=λ2(ε2σ2)dxdy,
with
λ1=1ε4(θ1η0κ+Z0θ1Σ1κ)andλ2=1ε4θ2Σ1κ,
and by neglecting the constant term (1/ε2)Σ1z2.
In doing so, the obtained scaled Hamiltonian (28) is nonsingular O(1) for ε → 0, that is,
H=[12Σ1|v1|2+12σ2|v2|2+(σ1+σ2)z2+12κθ1Σ1κ1σ12+12κθ2Σ1κ1(σ22+2σ1σ2)]dxdy.
The expression above is obtained by computing the Taylor expansion of (28) around (σ1=0,σ2 = 0) and by retaining only the terms at leading order in ε, with ε → 0.

APPENDIX B

Change of Variables in the Functional Derivatives

In this appendix we show how to compute the functional derivatives with respect to the initial variables (σ1, σ2, v1, v2) in terms of those derived in the asymptotic analysis (D1, ω1, φ1, σ2, v2). We start from the definition of the differential δF written in terms of both set of variables:
δF=[δFδσ1δσ1+δFδσ2δσ2+δFδv1δv1+δFδv2δv2]dxdy;
δF=[δFδD1δD1+δFδω1δω1+δFδϕ1δϕ1+δFδσ2δσ2+δFδv2δv2]dxdy.
By exploiting the relationships between one set of variables and the other, we can equate the terms in (C1) and (C2) that depend on the same differential. For example, since both D1 and ω1 are functions of v1, we can write
(δFδv1δv1)dxdy=(δFδD1δD1+δFδω1δω1)dxdy=[·(Σ1δFδD1δv1)Σ1δv1δFδD1+·(δFδω1δv1)δv1δFδω1]dxdy=[(Σ1δFδD1δFδω1)δv1]dxdy,
in which the divergence terms are zero due to the boundary conditions. In the expression above the differentials δD1 and δω1 have been computed as
δD1=δ[·(Σ1v1)]=·(Σ1δv1),
δω1=δ(·v1)=·δv1·

The functional derivatives with respect to the other variables can be obtained accordingly.

APPENDIX C

Scaled Variational Hamiltonian in the Limit ε → 0

Taking the variation of (B5) one obtains
δH=[Σ1v1·δv1+σ2v2·δv2+12|v2|2δσ2+(δσ1+δσ2)z2+κθ1Σ1κ1σ1δσ1+κθ2Σ1κ1(σ2δσ2+σ1δσ2+σ2δσ1)]dxdy=[Σ1v1·δv1+σ2v2·δv2+M1|ε=0δσ1+(12|v2|2+M2|ε=0)δσ2]dxdy,
in which Eq. (18) have been used. The first term in the integral above (a velocity) can be rewritten in terms of a potential χ and a streamfunction Ψ, i.e., Σ1v1=Σ1χ+Ψ, and subsequently manipulated as follows (using common vector calculus identities):
[Σ1v1·δv1]dxdy=[(Σ1χ+Ψ)δv1]dxdy=[Σ1χδv1]dxdy+[Ψδv1]dxdy=[(Σ1χδv1)χ(Σ1δv1)]dxdy+[(Ψδv1)Ψδv1]dxdy=[(Σ1χδv1)χδD1]dxdy+[(Ψδv1)Ψδω1]dxdy=[χδD1Ψδω1]dxdy
in which the divergence terms are zero due to the boundary conditions and the definitions of the differentials δD1 and δω1 in (B4) have been used.

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